CBSE Class 8 Mathematics Cubes and Cube Roots Notes Set 15

Cubes and Cube Roots

The cube of a number is the number multiplied by itself three times (n^3). A perfect cube is the cube of a natural number: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, ... Cubes of even numbers are even, cubes of odd numbers are odd and cubes of negative numbers are negative.
The cube root of a is the number whose cube is a, written as the cube root of a or a^(1/3). Cube root of 27 = 3 and cube root of -27 = -3.

Finding Cube Roots

Prime factorisation: write the number as a product of primes, group them in triples and take one prime from each triple. For 13824 = 2^9 x 3^3 the cube root is 2^3 x 3 = 24.
By inspection (estimation) for perfect cubes: the unit digit of the root follows from the unit digit of the cube (1, 4, 5, 6, 9 and 0 keep their digit; 2 and 8 swap; 3 and 7 swap). Cut off the last three digits; the remaining part lies between two cubes and gives the tens digit. For 46656: last digit 6 so the root ends in 6; 46 lies between 27 and 64 so the tens digit is 3; the root is 36.

Making a Number a Perfect Cube

Factorise the number; if some primes do not occur in a multiple of three, multiply by the missing primes to complete the triples, or divide by the extra primes. Example: 1029 = 3 x 7^3, so multiply by 3^2 = 9 to get 9261 = 21^3.

Digits and Sums

Digits in a cube root: put a dot over the unit digit and over every third digit to its left; the number of dots is the number of digits in the root.
1 + 2 + ... + n = n(n + 1)/2; 1^2 + ... + n^2 = n(n + 1)(2n + 1)/6; 1^3 + ... + n^3 = (n(n + 1)/2)^2.